# Tagged Questions

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### Know a formula for $x \pmod{y + z}$?

It seems natural to want to add the $x \pmod y$ operator to the integers (or other generalization of the integers?), but perhaps it has no algebraic properties. Can you prove that there is no ...
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### prove that if $a$ is not a unit in $\mathbb Z$/m$\mathbb Z$ then $a$ is a zero divisor in $\mathbb Z$/m$\mathbb Z$?

Ho can I prove that if $a$ is not a unit in $\mathbb Z$/m$\mathbb Z$ then $a$ is a zero divisor in $\mathbb Z$/m$\mathbb Z$ ? I am stuck on this problem I would appreciate a lot your help thanks!!
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### Show: $\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$ is a ring isomorphism for $m$ and $n$ relatively prim

Let $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Let $m,n\in\mathbb{N}$ be relatively ...
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### Show that the solutions lifted by Hensel's lemma, are the only solutions in $\mathbb{Z} / p^k \mathbb{Z}$

Suppose that $f(x) \in \mathbb{Z}[X]$ is a polynomial to which we can apply Hensel's Lemma. Suppose the set $A$ is the set of solutions of $f(x)$ in $\mathbb{Z} / p \mathbb{Z}$. I'd like to show that ...
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### Proposition 17, p. 68 of Lang's Algebraic Number Theory

I'm stuck on a detail of the following propositon: Let $K, E$ be linearly disjoint number fields with degrees $n$ and $m$ over $\mathbb{Q}$ whose discriminants (over $\mathbb{Q}$) are relatively ...
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### Let $R = \mathbb Z[i]$. Show $I \cap \mathbb Z$ is an ideal in $\mathbb Z$, for all $a \in I \cap \mathbb Z$, $10 \mid a^2 = N(a)$.

Let $R = \mathbb Z[i]$, $z = 3+i$ and $I = \langle z \rangle$. I need to show $I \cap \mathbb Z$ is an ideal in $\mathbb Z$, for all $a \in I \cap \mathbb Z$, $10 \mid a^2 = N(a)$ and $10 \mid a$, ...
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### Finding an $i \in \mathbb{N}$ s.t. $im + k = p$

Let $m,k \in \mathbb{N}$ s.t. $\gcd(m,k) = 1$. Let $\pi$ be a prime positive integer. Question: Does there always exist an $i \in \mathbb{N}$ s.t. $im + k = p$ s.t. $p$ is a prime positive integer ...
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### Halving One in Odd Size Rings

Consider the rings $\mathbb{Z} /n \mathbb{Z}$ where $n$ is odd. Every number is even in such rings. Assume we start with $1$ and keep "halving" until we get back to $1$. What can be said about the ...
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### Greatest common divisor of $2 + 3i$ and $1-i$ in $\mathbb{Z}[i]$

Greatest common divisor of $2 + 3i$ and $1-i$ in $\mathbb{Z}[i]$ Here is my attempt at solving this using a generalized Euclid's Algorithm. Does it look alright? Step 1 $2 + 3i = M(1-i) + N$ ...
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### Divisors of $x^3 + x + 1$ in $Z_3[x]$

Divisors of $f(x) = x^3 + x + 1$ in $Z_3[x]$ Do I have to manually check whether each polynomial in $Z_3[x]$ with degree less than $3$ divides $f$ or is there a better way? That's $3^3 = 27$ ...
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### Number of ideals in $\Bbb Z[x]/(x^3+1, 7)$

I am trying to find the number of ideals in $R:=\Bbb Z[x]/(x^3+1, 7)$ and $S:=\Bbb Z[x]/(x^3+1, 3)$. I started with $R$ and tried to write it in terms of familiar rings, by using fundamental ...
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### $(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$ [duplicate]

I'm trying to prove the group isomorphism $(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$. Obviously I tried to establish a ring isomorphism from $\Bbb Z[x]/(x^{n+1})$ ...
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### Idempotent elements in $(\mathbb{Z}_n,+,\cdot)$ [duplicate]

Can we find the idempotents in $(\mathbb{Z}_n,+,\cdot)$ for any $n$? Is there a general rule? Note: Trying to consider the prime factors!
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### In $ℤ/Nℤ$, which units are successors to zero divisors?

What are the units $x$ in $ℤ/Nℤ$ of the form $x = 1 + \overline{kd}$ for a divisor $d$ of $N$ and $k ∈ ℤ$, i.e. U_N[d] := \{x ∈ (ℤ/Nℤ)^×;\; ∃ k ∈ ℤ : x = 1 + \overline{kd}\} = \ker \big((ℤ/Nℤ)^× → ...
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### an invertible element $i$ in $\mathbb Z_n$ must be coprime to $n$

Let $n$ be an integer and $i\in \{1,\cdots,n-1\}$. I want to show that $i$ is invertible in $\mathbb Z_n$ if and only if $i$ is coprime to $n$. One way is easy. suppose $i$ is coprime to $n$ then ...
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### no prime numbers in a disc in $\mathbb{C}$ with radius R in $\mathbb{Z}[i]$

Show that there is a disc in $\mathbb{C}$ with radius R , so that no primes of $\mathbb{Z}[i]$ are contained in the disc. I was thinking of taking a disc which which does not touch (0,0), for ...
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### Modular Arithmetic question, possibly involving Chinese remainder theorem

'6 professors begin courses of lectures on Monday, Tuesday, Wednesday, Thursday, Friday and Saturday, and announce their intentions of lecturing at intervals of 2,3,4,1,6,5 days respectively. The ...
### Show $\mathrm{E}(\mathbb{Z}_{mn})$ is isomorphic to $\mathrm{E}(\mathbb{Z}_{m})\times\mathrm{E}(\mathbb{Z}_{n})$ if and only if $(m,n)=1$
Define $\mathrm{E}(\mathbb{Z}_{i})$ to be the group of invertible elements of the ring with unity $\mathbb{Z}_{i}$. Show that $\mathrm{E}(\mathbb{Z}_{mn})$ is isomorphic to ...
Let $a$ and $b$ be elements of the polynomial ring $\mathbb{Z}/n\mathbb{Z}[x]$. If $a$ and $b$ generate the same ideal, must they be associates? If $n$ is prime, then it is easy to see that the answer ...